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Городской семинар по теории вероятностей и математической статистике



On the defect ("signed area") of toral Laplace eigenfunctions and exponential sums Игорь Вигман^{} 

Аннотация: Язык доклада – русский This talk is based on a joint work with P. Kurlberg and N. Yesha. The defect (also known as "signed area") of a realvalued function defined on a twodimensional domain is the difference between its positive and negative regions. We are interested in the defect of toral Laplace eigenfunctions (exponential sums) restricted to Planckscale shrinking subdomains ("shrinking balls"). It is proved that, under a flatness assumption on the exponential sums, the defect asymptotically vanishes on the set of balls centres of almost full measure, for a generic sequence of energy levels. To establish our results we start from Bourgain's derandomization technique, borrow the IntegralGeometric sandwich from NazarovSodin, and also invoke other techniques. 